Signals are often processed to extract certain events from raw received signal data. Extraction of events from raw signal data requires that the events be detected within the raw signal data. Examples of events that may be present in received signals include repetitive synchronized events such as phase transitions in a phase-shift key (PSK) signal, frequency slope changes in a frequency modulated continuous wave (FMCW) signal, and frequency transitions in a frequency-shift key (FSK) signal. Such events occur with a frequency or a sub-frequency of a given baud. Other examples of repetitive synchronized events are envelopes of a PSK, FSK or FMCW signal that repeat in a non-random manner with a group pattern.
One existing real-time continuous wave (CW)/PSK system attempts to recognize element length for a PSK signal by collecting statistics of the time intervals between successive spikes from the rectified average delta phase derived from the PSK signal. Using this method, the most often occurring interval is recognized as the element length. However, this method is adversely affected by unwanted noise spikes that occur in-between good signal spikes, because such noise spikes change the time interval evaluation. For FMCW signals, the Kalman filter followed by a frequency peak detection method may be used to detect time intervals between successive frequency slope changes, and as with processing of PSK signals, the most often occurring time interval is recognized as the element length. However, this method is also adversely affected by unwanted noise created frequency slope changes that occur between two good frequency slope changes.
The Radon transform is known for its use in reconstructing images from medical computer tomography. In this application, the Radon transform describes the absorption of X-ray radiation as it traverses in a straight line in the human body. A formula for the Radon transform is:
  A  =            ∫      S        ⁢                  μ        ⁡                  (          x          )                    ⁢              ⅆ        x            
where A is the relative X-ray transmission, μ is the absorption coefficient, and the integral is taken along a straight line s. By inverting the above integral equation, an image of the absorption coefficient μ is constructed.
A “tau-p” transform is a form of the Radon transform used in seismic signal processing for attenuating straight line events, like the undesirable direct arrival. In a simple form, the Radon transform sums data values that lie on a straight line, and events that have the characteristics of a straight line can be identified by a large Radon sum. Because most noise is incoherent, it will not line up nicely in a straight line. Therefore the Radon transform is a useful tool in identifying straight-line coherent noise events like the direct arrival from a noisy background. In the case of seismic processing, the coherent direct arrival is identified and removed. Alternatively, the Radon transform may be used to identify and then extract lines, edges, curves, textures, or shapes. The Radon transform may be generalized to an integral along a curve, in which case it will enhance a curved event. The Hough transform is sometimes used where patterns are extracted from an image.
Time-time plots have been used to display climate study data reflecting intensity of a dry season and to display pulsed signal (i.e., radar) information.